1. Introduction
In remote areas such as small islands, diesel generators are the main source of power supply. Diesel fuel has several drawbacks: it is expensive because transportation to remote areas adds extra cost, and causes air pollution by engine exhaust. Providing a feasible economical and environmental solution to diesel generators is important. A hybrid system of wind power and diesel generators can benefit islands or other isolated communities and increase fuel savings. Wind is, however, a natural energy source that produces a fluctuating power output. The excessive fluctuation of power output adversely affects the quality of power in the distribution system, particularly frequency and voltage [1, 2].
A hybrid generation system is typically composed of a wind turbine coupled with an induction generator, a dieselsynchronous generator unit, a dumpload, and an energy storage system. In the operating with diesel-synchronous generator unit, the system voltage and frequency are controlled well by excitation and governor systems of the synchronous generator. However, it is problematic for frequency control when diesel-synchronous unit is not in operation since load perturbation in a completely isolated power system has a considerable effect on the system frequency. The load perturbation must be dealt with carefully in order to maintain frequency within the allowable range.
To overcome such problem, it is necessary to make use of an energy-storage unit with dumpload. The frequency regulation is accomplished by discharging the energy stored in batteries into the power network whenever there is a sudden frequency drop and charging them when the frequency increases sharply. Battery facilities are well suited for this task because it can provide fast active power compensation. Moreover, the dumpload consumes excessive network power when the battery is fully charged.
In this paper, the frequency is controlled by charging / discharging the battery storage system (BSS) and by absorbing the excess active power from the network with the dumpload, while the excitation system in the synchronous generator is used for the voltage control. If the battery storage is fully charged, the excess power must be absorbed by the dumpload. If the battery storage is not fully charged, the excess power will be charged into the battery and also absorbed by the dumpload. In this case, both the dumpload and the battery storage system operate since the battery is subject to the charging speed.
Some publications are concerned with energy storage system and dumpload combined with a wind turbine in remote areas [3-6]. These works do not provide detailed dynamic models of the battery storage system and the dumpload for stability analysis. In this paper, a nonlinear mathematical model of a hybrid wind system is developed for simulation purpose and for designing more effective controllers for power quality improvement. The nonlinear model of a hybrid wind system is formulated in the form of linear state space model with system matrices containing state variables. Thus, the use of robust control method for linear systems can be made for the developed model, where the state dependency of the system matrix is considered as the model uncertainty.
Model predictive control (MPC) is a kind of a finite horizon optimal control implemented in the form of a receding horizon control. Since it provides with closedloop stability and can effectively deal with constraints on the input or state, MPC has received much attention from both academia and industry [7-10]. In this paper, we adopt a simple Robust Tracking Controller (RTC) [10] to obtain a stabilizing tracking controller. The problem of obtaining robustly stabilizing gains of the controller is formulated in the form of linear matrix inequalities (LMI) so that it can be solved efficiently using software tools. The robust gain of RTC can be used to define a feasible and invariant set around a reference state and additional degrees of freedom can be adopted to enhance the performance and to enlarge the stabilizing region following the receding horizon strategy of MPC as it was done in [10]. But it would be complex and requires heavy computational burden.
This paper is organized as follows: the system model is examined in Section 2, with overall dynamic model given in the Appendix. In Section 3, three sub-models and the robust tracking controllers are developed. In Section 4, simulations are conducted, and conclusions are finally drawn in Section 5.
2. System Model
2.1 System configuration
The hybrid wind system consists of the wind turbine having the induction generator (IG), the diesel engine (DE) with synchronous generator (SG), a battery bank connected with a three-phase thyristor-bridge controlled converter, a dumpload, and the load. A three-phase dumpload is used with each phase consisting of seven transistor-controlled resistor banks. When wind generated power is sufficient to serve the load, the DE is disconnected from the SG by electromagnetic clutch, and the synchronous generator acts as a synchronous condenser.
The main purpose of the dumpload and the battery storage system (BSS) is to regulate the system frequency. The SG (with/without diesel) is used for reactive power control that is achieved by the excitation system to regulate voltage. The SG also contributes the reactive power to compensate for the induction generator. A current source converter is used for the BSS because the charging current can be critical to the battery life. A smooth charging can be achieved using a large inductor in the DC bus minimizing the current fluctuation. Moreover, current source converter is simpler than a voltage source converter.
Fig. 1 shows the overall configuration of the hybrid wind system [11]: Ca is the capacitor bank, Qd is the fuel flow rate at the governor chamber valve, Efd is the excitation field voltage, f is the frequency, Vb is the bus voltage, Lfilt is the AC side filter inductance, Cfilt is the AC side filter capacitance, Vc is the AC side voltage of the converter, and PBS is the battery power. Pdump is the dumpload power, and rdump is the dumpload resistance.
Fig. 1.The overall control action of hybrid wind system
2.2 Components models
The models of the generators are based on the standard Park’s transformation [12] that transforms all stator variables to the rotor reference frame described by a direct and quadrature (d-q) axis. The set of SG and IG equations are based on the d-q axis in accordance with International Electrotechnical Commission [13].
The nonlinear mathematical model of the hybrid wind system is derived and given in detail in [11] and [14]. The following considerations are taken into account to identify component models: the electrical system is assumed as a perfectly balanced three-phase system with pure sinusoidal voltage and frequency. High frequency transients in stator variables are neglected, which indicates that the stator voltage and currents are allowed to change instantly, because this paper is focused on the transient period instead of sub-transient period. Damper-winding models are ignored because their effect appears mainly in a gridconnected system or a system with several synchronous generators running in parallel. Different component models are of equal level of complexity.
The wind turbine operates in parallel with the battery storage system (BSS) through the current source converter, with or without the controllable dumpload. In either mode of operation, the diesel engine is disengaged by the clutch, and the synchronous generator is used for the voltage control. The active power is controlled either by the converter or the dumpload. In this section, the reduced-order model for the BSS and the dumpload along with the wind turbine is presented.
The algebraic electrical system equation from Appendix is as follows:
where
and electrical variables are:
The 10×10 electrical system matrix Λsys is defined in Appendix. The electrical variables in Vsys represent inputs to the 9th-order dynamic model that combines the windbattery storage system with the dumpload system.
3. Robust Tracking Controller Design
3.1 Control structure
Fig. 2 depicts the input and output relationship of the hybrid wind system from the control point of view. The control inputs are the excitation field voltage (u1) of the SG, the fuel flow rate at the governor chamber valve (u2), the battery power (u3), and the dumpload power (u4). The measurements are the voltage amplitude (y1) and the frequency (y2) of the AC bus. The wind speed (v1) and the load (v2) are considered as disturbances. From the control point of view, this model is a coupled 4 × 2 multi-inputmulti- output nonlinear system, since every input controls more than one output and every output is controlled by more than one input.
Fig. 2.The structure of the hybrid wind control system
3.2 Reduced-order model
Since the nonlinear model presented in [11] is too complex to design controllers, there is a need for a reduced-order model that is derived based on practical reasons; i.e., what is measurable and what can be manipulated. With the considerations in section 2.2, the reduced-order model assumes that the dynamic response of the converter is much faster than the desired bandwidth of the controlled system. This implies that the differential equations of the converter are neglected. Also, there is no elasticity in the drive train. Electrical dynamics of the induction generator is not explicitly modeled.
In deriving the nonlinear reduced-order models for the wind-BSS-dumpload system, the same simplifications applied to the previous section are used. In addition, it assumes that the dynamic response of the converter is much faster than the desired bandwidth of the controlled system, which implies that the differential equations of the converter are neglected. Then, the reduced-order model can be represented by the field flux linkage and the angular speed of the SG:
The control inputs are the field voltage (Efd) and either the dumpload power (Pdump) and the battery power (PBS). The outputs are the voltage (Vb) and the frequency (f= ωs). Eq. (3) needs a modification because the second control input does not explicitly appear in this equation.
The air gap torque of the synchronous generator Ts can be represented as
where PBS, Pdump, Ps, and Pind are the power of the battery, the dumpload, the synchronous generator, and the induction generator, respectively.
The reduced-order model becomes
With the same procedure as previous section, the final nonlinear reduced-order model is derived in the state-space form as
where
Note that the reduced-order model (6) is in the linear form for fixed system matrices Ac, Bc and Cc. However, matrices Ac and Bc are not fixed, but change as functions of state variables, thus making the model nonlinear. It should be advised that the reduced-order model is only the purpose of the designing controller, not for overall simulation study. The overall simulation is based on the model given in Appendix.
The RTC model represents a nonlinear system by partitioning the system into sub-systems. Three linear subsystems are considered for the nonlinear state-space model in (6) as
where Aci ∈ℜ2×2 , Bci ∈ℜ2×3 and Cci ∈ℜ2×2.
Here, the sub-systems are obtained by partitioning the state-space into three ranges of low, medium, and high levels for output variables Vb and ωs according to load variations. For each sub-space, different model (i=1, 2, 3) for possible low, the most possible, and possible high cases is applied according to the frequency (ωs) and voltage (Vb) variation. The continuous-time model (6) can be discretized with sampling period h as
where x[k]∈ℜ2 , u[k]∈ℜ3 , and y[k]∈ℜ2 are the states, inputs, and outputs, respectively, and the matrices , and C are obtained as follows
The objective of the control design is to regulate the output y[k] to the reference signal (thereby y[k]→r ). Considering the fact that Ac and Bc change as functions of state variables, we will assume that the system matrices and of (9) belong to the following polytopic uncertainty set:
where Ai = eAcih and
This kind of polyhedral type uncertainties are considered in the works of [15-17].
3.3 Robust tracking controller
In this paper, a robust control law is adopted for the discrete-time model (9) and (10) [10]:
Consider the control law
Note of (11) is the integration of the tracking error, y(k) − r. Thus, the proposed control will remove the steady-state error for a constant r if it stabilizes the closedloop system. In this paper, we will provide a method to determine the gains K and L such that the closed-loop system becomes stable in the presence of the uncertainty described as Ω . Here we consider a constant reference value r and its corresponding reference state x0 and reference control u0 satisfying the following relations:
Then, subtracting (12) from (9) yields the error dynamics:
where ex[k] := x[k]− x0 and eu[k] := u[k]− u0 .
Another error dynamics can be obtained as follows by subtracting the steady state value of from the both sides of the first equation of (11):
where . Combining (13) and (14), we have:
where and Ft = [K L],
Then, (15) can be rewritten as:
and the problem of finding stabilizing gain boils down to a standard state-feedback stabilization problem. Therefore, the gains stabilizing (16) can be obtained using the well known feedback design methods [15] as follows.
Consider a Lyapunov candidate function
Then, it is easy to see that V[k] > V[k +1] is guaranteed if
is met for all possible values of and . Using the Schur-complement [18], (18) can be transformed into the following LMI:
where R = Q−1 and . Since the matrix set (A, B) belongs to the uncertainty set Ω of (10), (19) can be guaranteed if it is met at every corners of Ω . Thus, the origin of the closed-loop system (16) is asymptotically stable in the presence of the uncertainty Ω if there exist a matrices R = Q−1 > 0 and Y such that
where and the feedback gains are determined as . The choice of R and Y satisfying the LMI (20), however, is not unique and we have to define an optimization problem to select one set of matrices R and Y. In order to make V[k] decreases as soon as possible in the presence of uncertainties, we can define an optimization problem as follows
and
where λ(R) is the generalized eigenvalue of R. Note that to control gain Ft becomes large when R becomes small and the additional constraint (22) is used to constrain the magnitude of control gains. The problem (21) can be solved effectively using a semi-definite programming. Note that the control (14) can be modified to yield the incremental formulation by subtracting u[k −1] from u[k] as follows:
This formulation helps to obtain a bumpless switching between manual or open-loop mode to closed-loop mode.
4. Evaluation by Simulation
4.1 System parameters
The system under study consists of a horizontal axis, 3-bladed, stall regulated wind turbine with a rotor of 16.6 m diameter, that runs an induction generator (IG) rated at 55 kW. The IG is connected to an AC bus in parallel with a diesel-synchronous generator unit that consists of a 50 kW turbocharged diesel engine (DE) driving a 55 kVA brushless synchronous generator (SG). Nominal system frequency is 50 Hz, and the rated line AC voltage is 230 V [19]. The battery storage is connected to the AC bus through a thyristor-bridge controlled current source converter rated at 55 kW. A load is rated at 40 kW. The inertia of the IG is 1.40 kgm2, and the inertia of the SG is 1.11 kgm2. The three-phase dumpload is used where each phase consists of 7 transistor-controlled resistor banks with binary resistor sizing in order to minimize quantum effects and provide near linear esolution.
The controller design parameters for the PI controllers of the governor, the excitation system, the converter, and the dumpload are set with the proportional gain 30 and the integral gain 90. The time-step size for overall simulation is 1ms.
Three linear models for possible low, the most possible, and possible high cases are obtained from (8) applying L=0.9 p.u., M=1.0 p.u., and H=1.1 p.u. for both Vb and f.
The control gain K and L were determined by solving the (18) and (20):
4.2 Hybrid wind power system control
This simulation is to examine how the battery storage system (BSS) and the dumpload work for better frequency control, and how they contribute the voltage control with the excitation system in the SG. Fig. 3 shows the wind speed, which can be described by a Weibull probability distribution. While the DG is shutdown, the load is changed from 38 kW to 20 kW at 5 sec. During the transient period from 5 sec. to 6 sec., the battery bank is charged with the speed of 700 W/sec. In the steady state, the battery is charged in 2 W/sec. Fig. 4 and Fig. 5 show the comparison of the active power of the IG, load, the dumpload, and the battery storage. It is observed that the proposed RTC has much faster response to the disturbance compared to the PI control.
Fig. 3.Wind speed in the charge operation
Fig. 4.Load change and power output of the SG
Fig. 5.Power outputs of the IG, the dumpload, and the battery storage
Fig. 6 and Fig. 7 show the comparison of the responses of the frequency and the voltage. Compared to the PI control, the frequency response of the RTC is excellent except for a slight bias from the nominal frequency, while the voltage response is sluggish compared to the PI control. From the simulation study, the proposed controller achieves the smoother and tighter power quality control in terms of the frequency, wind generator power, diesel generator power, battery power and dumpload.
Fig. 6.Bus voltage
Fig. 7.Bus frequency
The sluggish response of the voltage suggests that there is a room for further improvement in the robust tracking control. For a future work, a robust tracking model predictive control (MPC) is planned to enhance the robust performance in the presence of uncertainties of the reduced and discretized model as well as the uncertainties of the wind speed.
5. Conclusion
In this paper, the robust tracking controller is presented for the study of the power quality of the hybrid wind power system. The derived simulation model including the reduced-order model can be applied for different hybrid wind power system configurations to study power quality control. The proposed control scheme provides more effective control for the system to achieve better power quality, which is demonstrated by smooth transition of frequency, wind turbine generator, diesel generator, battery charge and dumpload. Thus, the usefulness of the robust tracking control is demonstrated in this paper. For a future work, a robust tracking MPC will be designed to enhance the robust performance in the presence of uncertainties of the reduced and discretized model as well as the uncertainties of the wind speed. The key issue of the future work will be to reduce the computational burden.
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